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A103941 Number of unrooted loopless n-edge maps in the plane (planar with a distinguished outside face). 3
1, 1, 2, 6, 22, 103, 614, 3872, 26414, 186988, 1367976, 10254326, 78461338, 610598818, 4821248244, 38546510368, 311560875422, 2542507084588, 20925300483992, 173530381632724, 1448900079476152, 12172334379246523, 102833593763830038, 873187910184763024, 7449120536014301138 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..500

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

FORMULA

For n > 0, a(n) = (1/(2n))*[binomial(4n, n)/(3n+1) + Sum_{0<k<n, k|n} phi(n/k)*binomial(4k, k)+q(n)] where phi is the Euler function A000010, q(n)=0 if n is even and q(n)=binomial(2n, (n-1)/2) if n is odd.

MATHEMATICA

a[n_] := (1/(2n)) (Binomial[4n, n]/(3n+1) + Sum[Boole[0 < k < n] EulerPhi[ n/k] Binomial[4k, k], {k, Divisors[n]}] + q[n]);

q[n_] := If[EvenQ[n], 0, Binomial[2n, (n-1)/2]];

Array[a, 20] (* Jean-François Alcover, Sep 01 2019 *)

PROG

(PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, if(d<n, 1, 1/(3*n+1)) * eulerphi(n/d) * binomial(4*d, d)) + if(n%2, binomial(2*n, (n-1)/2)))/(2*n))} \\ Andrew Howroyd, Mar 28 2021

CROSSREFS

Cf. A002293, A006390, A103942, A000260, A005470.

Sequence in context: A345194 A251181 A309579 * A064643 A218531 A339280

Adjacent sequences: A103938 A103939 A103940 * A103942 A103943 A103944

KEYWORD

easy,nonn

AUTHOR

Valery A. Liskovets, Mar 17 2005

EXTENSIONS

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 28 2021

STATUS

approved

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Last modified January 29 23:01 EST 2023. Contains 359939 sequences. (Running on oeis4.)